Pricing Interest Rate Derivatives Using Black-Derman-Toy Short Rate Binomial Tree

Transcription

1 Pricing Interest Rate Derivatives Using Black-Derman-Toy Short Rate Binomial Tree Shing Hing Man 3th August, 0 Abstract This note describes how to construct a short rate binomial tree fitted to an initial spot rate curve and volatility of short rate, using the Black-Derman-Toy model. With the resulted binomial tree, pricing of some interest rate derivatives are discussed. Introduction This note describes how to generate a short rate binomial tree fitted to an initial spot rate curve and volailtity of short rate, using the Black-Derman-Toy (BDT) model. The resulted short rate tree is used to price the following credit derivatives. Coupon paying bond American, European call, put option on bond Callable bond Cap and Floor Forward rate agreement Swaption Caps and floors on a floating rate agreement Capped, floored and collared FRNs The materials in this document are based mainly on [, Chapter 8] and [, Chapter 8]. In general, to price an interest rate derivative using a short rate binomial tree, the first step is to construct a short rate binomial tree from a given spot rate curve, and volatility of spot rate or short rate. The present value of an interest rate derivative is the discounted expected value of the payoff under the risk neutral probabilities. The expectation is usually calculated by backward induction on the binomial tree, which is similar to how a vanilla European option is valued using a binomial tree (please see [4, Chapter 5]). There are several interest rate models that can be used to generate a short rate binomial tree. In this document, the Black-Derman-Toy model is used. Another interest rate model is the Ho-Lee model([3, Chapter 5]).

2 The remainder of this document is organised as follow. In section, an example on how to price a coupon paying bond for a given short rate binomial tree is given. Then in section 3, given an initial spot rate curve and volatilities of short rate, an algorithm to construct a short rate binomial tree using the BDT model is derived. Finally, in section 4, methods to price the interest rate derivatives listed earlier using a given short rate tree are described. An Example On Pricing A Coupon Paying Bond Using A Short Rate Binomial Tree The purpose of this section is to illustrate the common approach in valuing an interest rate derivatives using a given short rate binomial tree. The example below on valuing a coupon paying bond is taken from [, Section 8.3]. Consider a short rate binomial tree over the next 4 years (please see [, Chapter 8] for the derivation of this binomial tree). No of ups 4 5.5% 3 8.4% 7.90%.84%.83%.74% 8.4% 8.78% 8,96% 9,07% 0 5% 5.64% 6.0% 6,5% 3 4 Time in years 6.46% At node (3, ), the short rate is 8.96%. At each node, the risk neutral probability of going up or down are both 0.5 ([, Section 8.]). Consider a 5-year bond starting at time 0, with notional $00 and a coupon of 0%, which is paid annually. Let V (t, j) be the value of the bond at node (t, j). Then at t = 5, the value of the bond is $0 ( notional + coupon). In other words, V (5, 0), V (5, ), V (5, ), V (5, 3), V (5, 4), V (5, 5) are all equal to $0. Then the value of the bond V (4, 0) at node(4,0) is given by V (4, 0) = (the discounted expected payoff at V (5, 0) and V (5, ) under the risk neutral probabilities) + the coupon at V (4, 0) ( = ) = 3.33 It is assumed that the short rate is also the spot rate. When the time step is small, the short rate is a good approximation of the spot rate.

3 Similary, V (4, ) = = 0.85 ( 0 + ) V (4, ), V (4, 3), V (4, 4) are worked out in the same way. Carry on like this, V (t, j) is obtained at t=3,,,0. The binomial tree below shows the value V (t, j) at each node (please see [, Table 8.3]). 5 No of ups V(t,j) Time in years The present value of the bond is V (0, 0) = Construction Of A Short Rate Binomial Tree For BDT Model This section gives a description on how to construct a short rate binomial tree for the Black- Derman-Toy (BDT) model, fitted to initial spot rate curve and short rate volatilities, using Arrow-Debreu prices (see Appendix). To start, lets define some notations. Let time steps t 0 = 0 < t < t <... be given. For i =,, 3,..., let D(t i ) be the discount factor over time period [0, t i ]. D(t i ) could be thought of as the value at t = 0 of a $ face value default free zero bond that matures at time t i. Note that D(0) =. Another version of BDT is to fit the initial spot rate curve and volatility of the spot rate. Please see [, Chapter 8, section 5]. 3

4 R(t i ) the interest rate over [0, t i ]. Note that simple interest is used. Thus D(t i ) = ( + R(t i )) ti σ(t i ) be the volatility, with respect to the risk neutral probability, of the short rate at time t i. D(t i, j) be the discount factor at time t i and state j, at (t i, j) for short, over the time period [t i, t i+ ]. r(t i, j) be the short rate at (t i, j). Note that when t i+ t i is small, r(t i, j) is a good approximation to the spot rate over [t i, t i+ ] at (t i, j). From now on, short rate also means spot rate. Define D(t i, j) = + r(t i, j) (t i+ t i ) Note that r(t 0, 0) = r(0, 0) is set to R(t ). (When t is small, R(t ) is a good approximation for r(0, 0)). At each time t i, it assumed without loss of generality that r(t i, j) will go up to r(t i+, j + ) with risk neutral probability. Hence r(t i, j) will go down to r(t i+, j) with risk neutral probability. r(t_{i}, j+) / r(t_{i }, j) / r(t_{i},j) Suppose for i, the following is satisified. Then for i, r(t i, j + ) = r(t i, j) e σ(ti) t i+ t i () D(t i, j) = + r(t i, 0)e jσ(t i) t i+ t i (t i+ t i ) Given D(t ), D(t ),..., D(t n ) and σ(t ), σ(t ),..., σ(t n ), where n, in the following, it is shown how to find r(t i, j) inductively,where i n, 0 j i, which satisfies () and there is no arbitrage opportunity. These r(t i, j) s are discretisation of the Black- Derman-Toy model d ln r = θ(t)dt + σ(t)dw Please consult [, Chapter 8, Section 4], [, Chapter 8] and [3, Chapter 5] for more details. At time t = 0, consider portfolio A that consists of a zero bond which matures at time t = t with a face value of $. () 4

5 portfolio B that consists of a derivative which pays { D(t, 0) at (, 0) D(t, ) at (, ) The value of portfolio A at time t = 0 is D(t ). The value of portfolio B at time t = 0 is G(t, 0)D(t, 0) + G(t, )D(t, ), where G(t, j) s are the Arrow-Debreu prices and they are known (see Appendix). As both portfolios have the same payoff at t = t, by the no arbitrage argument, their value at time t = 0 must be the same. Hence D(t ) = G(t, 0)D(t, 0) + G(t, )D(t, ) (3) From (), D(t, 0), D(t, ) could be expressed in terms of r(t, 0). Hence (3) would become an equation with one unknown r(t, 0). r(t, 0) could be solved using a numerical method. Once r(t, 0) is known, r(t, 0) follows from (). Now that the short rates at time t = t have been worked out, the short rates at time t = t is to be deduced next. At time t = 0, consider (new portfolios) portfolio A that consists of a zero bond which matures at time t = t 3 with a face value of $. portfolio B that consists of a derivative which pays D(t, 0) at (, 0) D(t, ) at (, ) D(t, ) at (, ) Both portfolios A and B have the same payoff at time t = t. argument they must have the same value at time t = 0. This gives By the no arbitrage D(t 3 ) = G(t, 0)D(t, 0) + G(t, )D(, ) + G(t, )D(t, ) (4) From (), D(t, 0), D(t, ), D(t, ) could be expressed in terms of r(t, 0). Hence (4) would become an equation with one unknown r(t, 0). r(t, 0) could be solved using a numerical method. Once r(t, 0) is known, r(t, ), r(t, ) follows from (). In general, suppose i 0 and r(t i, j) and G(t i, j) for j = 0,,..., i have been worked out. (Note that r(0, 0) = R(t ) and G(0, 0) =.) Then (see (7)) for j =, 0,..., i, G(t i+, j + ) = D(t i, j)g(t i, j) + D(t i, j + )G(t i, j + ) (5) The no arbitrage argument described above gives It follows from () that i+ D(t i+ ) = G(t i+, j)d(t i+, j) j=0 D(t i+ ) = i+ j=0 G(t i+, j) + r(t i+, 0)e jσ(t i+) t i+ t i+ (t i+ t i+ ) (6) 5

6 Note that (6) is an equation with one unknown r(t i+, 0). r(t i+, 0) could be solved using a numerical method (such as the Bisection method). Once r(t i+, 0) is known, the r(t i+, j) s, j =,,..., i +, could be deduced from (). 4 Pricing Interest Rate Derivatives Using A Short Rate Binomial Tree In this section, a way to price the following interest rate derivatives using a given short rate tree is described. Coupon paying bond American, European call, put option on bond Callable bond Cap and Floor Forward rate agreement Swaption Caps and floors on a floating rate agreement Capped, floored and collared FRNs In general, given a short rate binomial tree and the corresponding risk neutral probabilities, the initial price of an interest rate derivatives is the dsicounted expectation of the payoff under the risk neutral probabilities. The notation from Section 3 is kept. Also let p d (t i, j, ), p u (t i, j), be the risk neutral probabilities of going from (t i, j) to (t i+, j), (t i, j) to (t i+, j + ) respectively. In Section 3, in the construction of the BDT short rate binomial tree, p u (t i, j) d p d (t i, j) are set to / for all t i, j. From now on, it is assumed that p u (t i, j) and p d (t i, j) are set to /. For simplicity sake, the cashflows of the interest rate derivatives under consideration are at yearly intervals. This means the given short rate binomial tree is assumed implicitly to have time steps of one year in length. 4. Coupon Paying bond ([, p 500] ) Consider a bond with a face value of FV, which pays a coupon of r% at time t =,,...., T, where T is the maturity date. Let V (t, j) be the payoff (or value) of the bond at (t, j). Then V (T, j) = ( + r 00 )F V for j = 0,,..., T. The other V (t, j)s follow by backward induction. For a fixed t 0 >, suppose V (t 0 +, j) for all j = 0,,..., t 0 + are known. Then for j = 0,..., t 0, V (t 0, j) = (p d (t 0, j)v (t 0 +, j) + p u (t 0, j)v (t 0 +, j + ))D(t 0, j) + F V r 00 = V (t 0 +, j) + V (t 0 +, j + ) D(t 0, j) + F V r 00 6 (7) (8)

7 When t 0 = 0, V (t 0, 0) is given by ( ) V (, 0) + V (, ) V (0, 0) = D(0, 0) (9) The pricing of other interest rate derivatives are done in a similar fashion with minor modification to (7). 4. European Call and Put Option on bond ([, p 50] [, Section 8.6]) Consider a European call option on the bond described in Section 4. with (option) maturity date T option (where T option < T ) and strike price K. Let V (t, j) denote the value of the European call option. Then V (T option, j) = max(0, P (T option, j) K) for j = 0,,..., T option, where P (t, j) is the value of the bond at node (t, j). Suppose V (t 0 +, j) for all j = 0,,..., t 0 + has been worked out. Then for j = 0,..., t 0, V (t 0, j) = V (t 0 +, j) + V (t 0 +, j + ) D(t 0, j) (0) The price of a European put option on a bond could be derived in the same way, except that V (T option, j) = max(0, K P (T option, j)). Alternatively, if the value of the European call option is known, then the value of European put could be worked out from the Call-Put parity relation (see [3, p 477] ). 4.3 American Call and Put option on bond ([, p 50] [, Section 8.6]) Consider an American call option on the bond described in Section 4. with (option) maturity date T option. Let V (t, j) denote the value of the American call option. Then V (T option, j) = max(0, P (T option, j) K) for j = 0,,..., T option, where P (t, j) is the value of the bond at node (t, j). Suppose V (t 0 +, j) for all j = 0,,..., t 0 + has been worked out. Then for j = 0,..., t 0, V (t 0, j) = max(max(0, P (t 0, j) K), recurv (t 0, j)) () where recurv (t 0, j) = V (t 0+,j) + V (t 0 +,j+) D(t 0, j). For an American put option, set V (T option, j) = max(0, K P (T option, j)) for j = 0,,..., T option. 4.4 Callable Bond ([, p 50] ) Suppose the bond described in section 4. is a callable bond. It means the holder of this callable bond is the holder of a conventional bond and has written an American call option on the bond. The issuer of the bond holds this American call option. Recall that T is the maturity date of the conventional bond. The American call option expires at T. Hence the value of this callable bond is the value of the conventional bond the value of the American call option () The price of conventional bond and American call option on bond have already been discussed earlier. 7

8 4.5 Cap and Floor [, p 503] Consider an interest rate cap that matures at time T cap T with a strike rate of K% and notional principal NP. Upon on maturity the holder of this cap can choose to borrow (but no obligation) NP over [T cap, T cap + ] 3 at a rate of K%. Let V (t, j) be the payoff or value of this cap. Then, for j = 0,,..., T cap V (T cap, j) = max(0, r(t cap, j) K 00 ) NP D(T cap, j) (3) Suppose V (t 0 +, j) for all j = 0,,..., t 0 + has been worked out. Then for j = 0,..., t 0, V (t 0, j) = V (t 0 +, j) + V (t 0 +, j + ) D(t 0, j) (4) For an interest rate floor, the payoff at maturity date T floor < T is V (T floor, j) = max(0, K 00 r(t floor, j)) NP D(T floor, j) (5) (4) could be used to deduce the value of interest rate floor V (0, 0). 4.6 Forward Rate Agreement [, p 504] Let T F RA T. The holder of a T F RA (T F RA + ) 4 FRA with notional principal NP and forward rate r% must borrow NP over [T F RA, T F RA + ] at a rate of r% per time step. Let V (t, j) be the value of this FRA. Then, for j = 0,,..., T F RA V (T F RA, j) = (r(t F RA, j) r 00 ) NP D(T F RA, j) (6) Suppose V (t 0 +, j) for all j = 0,,..., t 0 + has been worked out. Then for j = 0,..., t 0, 4.7 Swaption V (t 0, j) = V (t 0 +, j) + V (t 0 +, j + ) D(t 0, j) (7) Consider a swap agreement that starts from T option and ends at T, with notional principal NP and fixed rate K%. The exchange of interest takes place on t = T option +, T option +,..., T. An option to buy the above swap at t = T option, with the holder of swap paying fixed rate payment is called a payer swaption. The above payer swaption has the same payoff as (see [, p 5] ) a put option on a bond that matures at T, pays a coupon rate of K% with a face value of NP 3 For simplicity sake, the loan period is assumed to be year. In general, the period could any length. 4 Again, for simplicity sake, the period is year. 8

9 with strike price NP and (option) maturity date T option. Also, any coupon payment from bond on T option is excluded in pricing the put option. The price of a put option on bond has already been discussed eariler. Similarly, a receiver swaption is where the holder of the swaption receives fixed rate and pays floating rate. A receiver swaption could be price as a call option on a bond. An alternative way to price a swaption is to work out the swap rate at each step at t = T option. From the swap rate, the payoff of swaption could be derived. The present value of the swaption is obtained by the usual backward induction from t = T option. Please see [, p506] for details. 4.8 Caps and floors on a floating rate note [, p 508] A floating rate note (FRN) on an agreed principal which matures at time T, pays floating interest on the principal at time t =,,..., T, and pays the principal at time T. The rate payable at time t is the rate at time t. More precisely, for i = 0,,..., T, let r i be interest rate at time i over [i, i + ]. Then, the interest rate payable at time t is r t, where t =,,..., T. It is well known that the value of an FRN at time t = 0,,,..., T is the principal (see [, Appendix 4.]). A capped FRN with a cap rate K cap % is a modified FRN with interest rate at time t being min(r t, K cap ). In other words, the interest rate is capped at K cap %. Let V cap F RN (t, j) be the value of a capped FRA with principal P and cap rate K cap %. Then, for j = 0,,..., T V cap F RN (T, j) = (Cash flow at time T ) D(T, j) (8) = P ( + min(r(t, j), K cap) ) D(T, j) (9) 00 (Note that r(i, j) and K cap are assumed to be in %.) The V cap F RN (t, j)s for t < T are derived by backward induction. Suppose V cap F RN (t +, j) for j = 0,,..., t + have been worked out. Then, for j = 0,,..., t, V cap F RN (t, j) = V recurs (t, j) + P min(r(t, j), K cap ) D(t, j) (0) where V recurs (t, j) = V cap F RN (t+,j)+v cap F RN (t+,j+) D(t, j). The cash flow of the capped FRN at (t, j) is made of cash flows at t +, t +,..., T and each of which is discounted accordingly. In (0), P min(r(t, j), K cap ) D(t, j) is the contribution from t+ and V recurs (t, j) is the contribution from t +, t + 3,..., T. A floored FRN with principal P, mature time T and floor rate K fl % is a modified FRN with interest rate payable at time t =,,... T being max(r t, K fl ). Let V fl F RN (t, j) be the value of of the above floored FRN. Similar to capped FRN, and for t < T, V fl F RN (T, j) = P ( + max(r(t, j), K fl) ) D(T, j) () 00 V fl F RN (t, j) = V recurs (t, j) + P max(r(t, j), K fl ) D(t, j) () where V recurs (t, j) = V fl F RN (t+,j)+v fl F RN (t+,j+) D(t, j). 9

10 A collared FRN with principal P, mature time T, capped rate K cap %, floored rate K fl % is defined in the obvious way. Let V coll F RN (t, j) be the value of the above collared FRN. Then (see [, page 509]) V coll F RN (t, j) = P V cap F RN (t, j) + V fl F RN (t, j) (3) Remark A short rate binomial tree could be generated by other models (eg Ho-Lee) other than BDT. Appendix Arrow-Debreu price Let r(t i, j) be the short rate at time t i and state j, at (t i, j) for short, over time period [t i, t i+ ] on a binomial tree. Let p be the risk neutral probability that the short rate will go up from r(t i, j) to r(t i+, j + ). (Hence r(t i, j) will go down to r(t i+, j) with probability p. p r(t_{i+}, j+) r(t_{i}, j) p r(t_{i+}, j) For integers 0 i 0 and 0 j 0, let G(t i0, j 0 ) be the value of a derivative at time 0 and the payoff at t = t i0 is given by δ j0j where j is the state reached at time t i0 (4) (G(t i0, j 0 ) also denotes the above defined derivative.) Note that G(0, 0) is. The G(t i, j) s are known as the Arrow-Debreu prices. Let V (t i, j) be the value (payoff) of an arbitrary derivative V at (t i, j). Let i be given. As V and i s=0 V (t i, s)g(t i, s) have the same payoff at t i, by the no arbitrage argument, they must have the same value at t = 0. Hence V (0, 0) = Let t i0, j 0 be given. The value of G(t i0, j 0 + ) at time t i0 is ( p)d(t i0, j 0 + ) at state j 0 + pd(t i0, j 0 ) at state j 0 0 otherwise i V (t i, s)g(t i, s) (5) where D(t i, j) is the discount factor at (t i, j) over [t i, t i+ ]. Then { e r(t i,j)(t i+ t i ) for continuous interest D(t i, j) = for simple interest s=0 (+r(t i,j)) (t i+ t i ) (6) 0

11 r(t_{i },j+) r(t_{i}, j+) r(t_{i },j) r(t_{i},j) Let i, j i be given. By (6), the payoff of G(t i, j + ) at (t i, s) is ( p)d(t i, j + ) if s = j + pd(t i, j) if s = j 0 otherwise Apply (5) with V = G(t i, j + ) at t = t i to get G(t i, j + ) = ( p)d(t i, j + )G(t i, j + ) + pd(t i, j)g(t i, j) (7) Note that by defining G(t i, j) = 0 if i < 0 or j < 0 or j > i, it follows from (7), that G(t i, j) could be calculated inductively. References [] L Clewlow and C Strickland Implementing Derivatives Models, Wiley [] K Cuthbertson and D Nitzsche, Financial Engineering-Derivatives and Risk Management, Wiley [3] R Jarrow and S Turnbull, Derivative Securities, South-Western College Publishing [4] Paul Wilmott, Introduces Quantitative Finance, Wiley 00